A cylindrical glass is half full of lemonade. The ratio of lemon juice to water in the lemonade is 1:11. If the glass is 6 inches tall and has a diameter of 2 inches, what is the volume of lemon juice in the glass? Express your answer as a decimal to the nearest hundredth.
We can begin by calculating the volume of the liquid in the glass.  Since the glass is half full, the portion filled with liquid has height 3 inches.  The volume will be $\pi r^2 h=\pi\cdot 1^2 \cdot 3 = 3\pi$.  Now, since the ratio of lemon juice to water is 1:11, the ratio of lemon juice to the liquid will be 1:(1+11) which is 1:12.  So, the volume of lemon juice in the glass is: $$3\pi \cdot \frac{1}{12}=\frac{\pi}{4}\approx .7854$$ So, the answer is $\boxed{.79}$ to the nearest hundredth.